Interactions of Classical and Numerical Algebraic Geometry

经典与数值代数几何的相互作用

• 批准号: 0756904
• 负责人: Daniel Bates
• 金额: $ 2.25万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-03-15 至 2009-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0756904&HistoricalAwards=false
• 关键词: Interactions; Classical; Numerical; Algebraic; Geometry

Algebraic geometry is a classical discipline which has for many years been situated at the intersection of algebra, number theory, several complex variables, and geometry in all its incarnations. The advent of personal computing, and more so the development of software for symbolic computation, introduced a new facet of the discipline; it was suddenly possible to carry out computations far too sophisticated to be handled manually. More recently, a numerical approach to algebraic geometry was pioneered, largely driven by the work of Sommese, Verschelde, and Wampler. This new field is aptly known as numerical algebraic geometry. The fundamental technique of this field, known as homotopy continuation, provides a numerical means of tracking the changing geometry resulting from algebraically morphing one system of polynomial equations into another. The ubiquity of polynomial systems throughout the sciences and engineering means that the techniques of numerical algebraic geome try may be used to solve problems arising in many different disciplines, such as kinematics, chemistry, economics, robot vision, and power electronics.It has recently been realized that the techniques of numerical algebraic geometry may be used to study problems in classical algebraic geometry as well. The purpose of this conference is to bring together classical and numerical algebraic geometers in a lively discussion forum to spark joint efforts between these two communities. The intersection of these two fields is ripe for rapid growth fueled by joint work, so it is the hope of the organizers that these interactions will help to yield important advances along this front. The conference will take place at the University of Notre Dame from May 22 to May 24, 2008. There will be 13 lectures over those three days, given by eminent scholars from both communities and from various locations throughout Europe and the U.S. As this field seems poised to spawn many new avenues of research, the participation of young mathematicians will be actively sought.

代数几何是一个经典的纪律，多年来一直位于交叉代数，数论，几个复杂的变量，并在其所有化身几何。 个人计算机的出现，尤其是符号计算软件的发展，为这门学科带来了一个新的方面;突然之间，人们可以进行过于复杂而无法手动处理的计算。 最近，代数几何的数值方法是开创性的，主要是由Sommese，Verschelde和Wampler的工作驱动的。 这个新领域被恰当地称为数值代数几何。 这个领域的基本技术，被称为同伦延拓，提供了一种数值方法来跟踪从一个多项式方程组代数变形到另一个多项式方程组所产生的几何变化。 多项式系统在科学和工程中的普遍存在意味着数值代数几何技术可以用来解决许多不同学科中出现的问题，如运动学、化学、经济学、机器人视觉和电力电子学。最近人们意识到，数值代数几何技术也可以用来研究经典代数几何中的问题。 这次会议的目的是汇集古典和数值代数几何学家在一个活跃的讨论论坛，以激发这两个社区之间的共同努力。 这两个领域的交叉点已经成熟，可以在联合工作的推动下迅速发展，因此组织者希望这些互动将有助于在这条战线上取得沿着的重要进展。 会议将于2008年5月22日至5月24日在圣母大学举行。 在这三天里，将有13场讲座，由来自两个社区和欧洲和美国各地的著名学者主讲，因为这一领域似乎有望产生许多新的研究途径，将积极寻求年轻数学家的参与。